Quantum Geometry and Its Rami¯cations Key feature of general relativity: gravity is encoded in ge- ometry. There are no background ¯elds |no background Riemannian geometry. The central idea in Loop Quantum Gravity is to elevate this interplay between gravity Riemannian geometry to the quan- tum level, preserving the background independence. Matter and Geometry, both quantum from `birth' ) Have to learn to do physics without a classical space-time geometry. Goal of this talk: Explain basics of quantum Riemannian geometry and illustrate its rami¯cations. In particular, the continuum breaks down near singularities and quan- tum geometry e®ects lead to new vistas. Only broad-brush Strokes. Mathematically oriented review: AA+ Lewandowski, CQG 2004; gr-qc/0404018 Organization: 1. Elements of Quantum Riemannian Geometry ² Underlying ideas ² Breakdown of the continuum 2. Quantum black holes ² Hints and challenges ² Quantum geometry and entropy ² Glimpses of the quantum space-time beyond singularity 1. Elements of Quantum Riemannian Geometry ² Geometry: Physical entity, as real as tables and chairs. Riemann 1854: GÄottingenAddress; Einstein 1915: General Relativity ² Matter has constituents. GEOMETRY?? `Atoms of Geometry'? Why then does the continuum picture work so well? Are there physical processes which con- vert Quanta of Geometry to Quanta of Matter and vice versa? ² A paradigm shift to address these issues \The major question for anyone doing research in this ¯eld is: Of which mathematical type are the variables . which permit the expression of physical properties of space. Only after that, which equations are satis¯ed by these variables?" Albert Einstein (1946); Autobiographical Notes. Choice in General Relativity: Metric, g¹º . Directly determines Riemannian geometry: Geometrodynamics. In all other interactions, by contrast, the basic variable is i a Connection, i.e., a matrix valued vector potential Aa; Gauge theories: Connection-dynamics Key New idea: `Kinematic uni¯cation'. Cast GR also as a theory of connections. Import into GR techniques from gauge theories that do not refer to a background geometry. Holonomies and Triads ² As in Wheeler's Geometrodynamics, Hamiltonian framework (a la Dirac): Good mathematical control for a non-perturbative, background independent treatment. [Path Integral Approach: Spin Foams] i a ² Phase space: ¡ 3 (Aa;Ei ) on a 3-manifold M. i a Aa: An SU(2) connection and Ei : a su(2)-valued vector den- sity on M; Z i a 3 ­ j(A;E) (±1; ±2) = (±1Aa)(±2Ei ) ¡ (±1 Ã! ±2) d x M Phase space (¡; ­) the same as in SU(2) YM theory. ² But Dual interpretation (SU(2) is the (double cover) of the group i of rotations of orthonormal triads on M. Aa parallel transports chiral spinors.) Due to this `soldering': Relation to geometrodynamics a bi ab Ei E = const q q a Ei » orthonormal triad (with density weight 1) i i i Aa = ¡a(E) ¡ (const)ka Classically, only a canonical transformation but can import new quantization techniques from gauge theories. ² `Elementary' Phase SpaceR Functions: ? Holonomies: he := P exp ¡ e A; Con¯guration variables: Ring R of functions on A generated by matrix elements of holonomies. R i a ? Momenta: Electric/Triad Fluxes: ES;f := S f Ei Via Poisson brackets, Fluxes de¯ne derivations on R: fF; ES;f g := LX(S;f)F; 8F 2 R. Situations completely parallel to QM on manifolds. Quantum Theory ² Construct the free ? algebra generated by elementary variables and mod by relations. The task is to construct suit- able representations of this algebra a. ² Surprising uniqueness result: `The quantum algebra a admits a unique irreducible di®(M) invariant (i.e., background independent) representation'. (Lewandowski, Okolow, Sahlmann, Thiemann; Fleischhack) Contrast with QFTs in Minkowski space-time. Background independence unexpectedly powerful. Representation (AA, Baez, Lewandowski, Rovelli, Smolin, ...) ² Quantum Con¯guration space A¹: Gel'fand spectrum of the Abelian unital C? algebra generated by the matrix ele- ments of holonomies. (A¹ is the compact topological space, the C? algebra of all continuous functions on which is isomorphic to the holonomy C? algebra.) ² `Controllable': A¹ consists of `generalized' connections A¹: A¹ associates to each edge e an element A¹(e) of SU(2) such ¡1 ¡1 that (i) A¹(e1) ¢ A¹(e2) = A¹(e1 ¢ e2); and (ii) A¹(e ) = [Ae¹ ] . 2 ¹ ² State Space: Hgeom = L (A; d¹o).L Spin-network decomposition: Hgeom = ®;~j H®;~j (Finite dim Hilbert spaces of spin systems.) This greatly facilitates calculations. `Typical states': Given a graph γ with n edges and a function n F on [SU(2)] , set ªγ;f (A¹) = F (A¹(e1);::: A¹(en)). By varying γ; F one obtains a dense subspace of Hgeom. ² Operators: Holonomies and (smeared) triads are self- adjoint on Hgeom . h^e act by multiplication (Gel'fand theory) E^S;f triads act by derivations. (Recall PBs de¯ne deriva- tions.) Quantum Riemannian geometry constructed from triad operators. Examples of Novel features ? All eigenvalues of geometric operators discrete. Eigen- values not just equally-spaced but crowd in a rather sophisti- cated way. Geometry quantized in a very speci¯c way. (Recall Hydrogen atom.) ? Fundamental excitations of geometry are 1-dimensional (holonomies). p`Polymer geometry'. Flux-lines of area, with evs 2 aj = const j(j + 1)`Pl. Continuum only a coarse-grained approximation; a fabric woven by quantum threads. ? Inherent non-commutativity: Areas of intersecting sur- faces don't commute. Inequivalent to (at least the naive) quantum geometrodynamics. ? No operator corresponding to the connection itself. Mathematical Analogy: U^(¸) := exp\i¸x well-de¯ned but not x^. In QM, the von-Neumann uniqueness theorem no longer applicable ) New Quantum mechanics possible. Realized in mini-superspace reductions, e.g., of quantum cosmology. 2. Quantum Black Holes ² First law of BH Mechanics + Hawking's discovery that 2 TBH = ·~=2¼ ) for large BHs, SBH = ahor=4`Pl ² 1. Entropy. Why is the entropy proportional to area and not volume? For a M¯ black hole, we must have exp 1076 micro-states, a HUGE number even by standards of statistical mechanics. Where do these micro-states come from? For gas in box, the microstates come from molecules; for a ferromagnet, from Heisenberg spins; Black hole ? Cannot be gravitons: gravitational ¯elds stationary. ² To answer these questions, must go beyond the classical space-time approximation used in the Hawking e®ect. Must take in to account the quantum nature of gravity. ² Distinct approaches. In Loop Quantum Gravity, this entropy arises from the huge number of microstates of the quantum horizon geometry. `Atoms' of geometry itself! Quantum Geometry and Entropy ² Heuristics: Wheeler's It from Bit: Divide the hori- 2 zon into elementary cells, each carrying area `Pl and as- sign to each cell a `Bit' i.e. 2 states. 2 n Then, # of cells n » ao=`Pl; No of states N» 2 ; S » 2 2 ln N» n ln 2 » ao=`Pl. Thus, S / ao=`Pl. ² Argument made rigorous in quantum geometry. Many inaccuracies of the heuristic argument have to be overcome. Interesting mathematical structures ( U(1) Chern- Simons theory; non-commutative torus, quantum U(1), ...) . Detailed treatment: (AA, Baez, Krasnov; Domagala, Lewandowski; Meissner; AA, Engle, Van Den Broeck, ...) ? Black holes in equilibrium modelled by Isolated Hori- zons. Only the horizon is isolated |external space-time can be dynamical. Laws of BH mechanics continue to hold. BHs can be rotating, distorted by external matter, hair, etc. ? Construct the phase space of general relativity consist- ing of space-times which admit an Isolated Horizon with a ¯xed set of multipoles as the inner boundary (a set of numbers which provide a di® invariant characterization of the classical horizon geometry.) ? Quantize this sector of GR + matter. In the bulk: states are polymer-like |described by Hgeom. But there are also surface states associated with the horizon. The two are inter- twined by the quantum isolated horizon boundary conditions. Physically, the quantum horizon can fluctuate and so can the polymer excitations. But they must do so in tandem. ? Microstates responsible for entropy arise from the in- trinsic geometry of the quantum isolated horizon. Described by a topological theory: U(1) Chern-Simons theory. p 2 ? Area of elementary `cells' const j(j + 1)`p; j can be an arbitrary 1/2 integer. Cells don't have the same area. ? Each cell has (2j + 1) states; not just 2. Final Result Careful counting of surface states (Domagala, Lewandowski, Meissner) shows that for large BHs, the leading contribution to the entropy is proportional to the horizon area as required by the Bekenstein-Hawking semi-classical formula. The sub- leading, quantum gravity correction has also been calculated (Meissner) . [For simplicity of presentation, have not discussed the pre- cise numerical coe±cients. Related to unspeci¯ed proportion- ality constants |Barbero-Immirzi parameter.] Quantum resolution of Singularities: First Steps ² The Big bang Singularity: Detailed, analytical and nu- merical treatment in mini-superspace models was discussed in this program. So, I will now consider the Schwarzschild singularity. (AA, Bojowald) ² Interior of the horizon of Schwarzschild solution is fo- liated by homogeneous slices M = R £ S2 with 4 Killing ¯elds. Kantowski-Sachs symmetry group. So, as a ¯rst step we can carry out a symmetry reduction and focus on the Kantowski-Sachs mini-superspace. Symmetry adapted coor- dinates (x; θ; Á); metric functions: (qxx; r). 2-d conf space and 4-d phase space. ² Connection dynamics: A = co¿3dx + c1¿2dθ ¡ c1¿1 sin θd' + ¿3 cos θd' E = po¿3 sin θ@=@x + p1¿2 sin θ@=@θ ¡ p1¿1@=@' ¡ 3 (co; c1; po; p1) ´ (¡co; c1; ¡po; p1) ? Orthonormal co-triad: (x) (x) (θ) (θ) (') (') ! = f dxp; ! = f dθ; ! = f sin θd'p where, (x) (θ) (') f = sgn po jp1=poj; f = ¡f = sgn p1 jpoj ? Scalar/Hamiltonian constraint: 2 C = ¡(poc1co + (c1=2 + 4)p1) = 0 ² Mini-superspace M 3 (po; p1) 40 m3=3 30 20 m2=2 10 m1=1 c p 0 -10 -20 -30 -40 0 1 2 3 pb ² Wheeler-DeWitt type quantization: 2 2 2 H = fª(po; p1) 2 L (R ; d x); ª(po; p1) = ª(po; ¡p1)g p^o; p^1 act by multiplication; c^o; c^1 act by di®erentiation.